Celebrating the Small Wins

Two weeks ago, after scoring an end of the first marking period test taken by two of my algebra 1 sections, thoughts of doom and despair entered my mind.  Students obviously struggled with the content, as they averaged a 38% on the test.  While they had seen nearly every problem either on an earlier quiz, homework assignment, in-class assignment, or lecture, and multiple “checks for understanding” indicated the majority (~80%) understood the concept or procedure, either they did not at the time, or their ability to recall and apply what they learned afterwards diminished significantly.  Six weeks of effort meant to establish a level foundation for my students did not yield the results I expected.  The enormity of the situation weighed heavily upon my mind.  I could not continue on to the next series of topics: inequalities.  Reteaching the content using the test as a guideline for my students seemed the only reasonable choice.

Clearing my mind of the devastation I felt,  I carefully reviewed each section of the test with my students re-emphasizing the concept and procedures specific to each.  Students had the initial test from which to review, and three full days reviewing the solutions both with me showing how, as well as various students who answered particular problems correctly.  With this review completed, students took a nearly identical test, with certain problems the same on the retake as on the initial test.

Unfortunately, when scoring the retakes, a similar feeling crept into the fore.  With nothing to substantiate the feeling other than the incorrect marks on each test, and the stack of scored tests showing total points lost overall and per page, my initial impression was that students did not improve on the retake.

After entering their results into Excel, analyzing them, and creating various charts, I found a reason to remain hopeful.  Yet, while they did improve, with an average score of 43%, I had hoped for more; the average score just barely topped the cut score I use to delineate a “far below basic” from a “below basic” level of understanding, which translates to an F and a D, respectively, come letter grade time.   Nonetheless, between the two sections, they improved the average by five percentage points, or twenty-two percent.  In magnitude, the 22% improvement sounds decent, however, as we started from such a low initial score, we remained at a low score after the retake.  Notwithstanding the fact that we still had a long way to go for the majority of the class to achieve a basic level of understanding, we did improve scores, which deserved a small celebration, at least in terms of reigniting my passion to improve my students’ understanding of algebra so that they pass the course, and remain on track to graduate high school.

I plan to rally my students by celebrating their success, and explaining the situation and plan forward.  At the same time, the sober fact that the average for the two sections remains nearly thirty percentage points from where I would like to be will keep the celebration short lived.  The plan right after is to return to square one for these concepts so that retake part deux achieves a more significant improvement.  This requires setting the pacing guide aside for a while, and likely even a revision to the guide.  Fortunately, I have the latitude to do so as I helped create the pacing guide, which is based on the Common Core State Standards.  I originally planned to teach to the Common Core pacing guide once I established a more level playing field (of understanding and procedural fluency) for my students.  The field is still quite uneven, so further review, reteaching, and relearning is required.

Skills and understanding intended to be established in elementary and middle school are holding these students back from attaining higher scores; surprisingly, over 90% of my algebra 1 students took algebra 1 in eighth grade, so these skills should have been re-established already.  But they have not.  Until the root cause for the various misunderstandings, misconceptions, or mistakes are identified and addressed, scores will remain stuck in the below basic and far below basic levels, which is insufficient for success with the richer content and process standards of the Common Core.

The following figures illustrate multiple aspects of the results.  As one can see, initial and retake scores are nowhere near proficient for more than a few students.

Correlation Between Retake and Initial Scores

As the figure shows, a significant number of students improved on the retake, while others worsened slightly.  The average gain shows clearly as the regression line nearly parallels the red line, which defines a line of perfect, positive correlation.

Initial and Retake Scores Per Student

Individual student scores ranged from exceedingly low to quite high.  Many students improved upon their initial score.  However, far too many, close to 80% of students, fall into the far below basic and below basic levels, which needs to be addressed before we move on with new content.  I believe it may require two or more weeks to remedy, and this assumes full student engagement; in other words, students must want to master these concepts before scores improve significantly, irrespective of the length, or intensity of the review.

Rating Distribution

This figure underscores the enormity of the challenge facing us.  Fortunately, there was positive movement in the three-day period, which I hope to build upon starting this coming Monday.

Score Distribution

Wrapping up the various ways to view results of the retake, it is nice to see the decrease in far below basic scores and the accompanying increase in basic and advanced scores.  Now, the challenge is to shift the entire distribution more to the right.

The Test

Test problems covered pre-algebra and beginning algebra topics spanning the following concepts.

Ordering Numbers:

Order of Operations:

Evaluating Expressions:

Writing and Evaluating Expressions:

Simplifying Expressions:

Solving (multistep) Equations [noting if any are an identity or a no solution]:

Solving Word Problems:

Distance / Rate / Time:

Dimensions of a Rectangle [given the perimeter and the relationship between length and width]:

Finding Consecutive (even / odd) Numbers [that sum to a number]:

Number Type – [Names, Symbols, and  3 Examples: Natural, Whole, Integer, Rational, and Irrational]:

About Dave aka Mr. Math Teacher

Independent consultant and junior college adjunct instructor. Former secondary math teacher who taught math intervention, algebra 1, geometry, accelerated algebra 2, precalculus, honors precalculus, AP Calculus AB, and AP Statistics. Prior to teaching, I spent 25 years in high tech in engineering, marketing, sales and business development roles in the satellite communications, GPS, semiconductor, and wireless industries. I am awed by the potential in our nation's youth and I hope to instill in them the passion to improve our world at local, state, national, and global levels.
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12 Responses to Celebrating the Small Wins

  1. Math Mom says:

    The test covers a lot of ground. Maybe it can be broken into 2 parts.

    Have you created any study tools for kids to do the nights prior to the test?

    I am not convinced that kids know how to study effectively on their own (I have a 12 year old).

    If they are self-checking their homework in school, it may very well still have wrong answers on it (again, I have a 12 year old).

    Their notes may not be correct or even readable.

    What has been helpful for my son is clear, type-written notes to review as his are often illegible.

    In cases with defitions (number systems) making a Quizlet study tool so kids can actively practice is a good option.

    Practice makes perfect. Kids won’t create their own math practice. It must be provided.

    I think the more you can guide your students (and make sure they are getting thenwork done correctly before the test) the better results you will have


    • Thanks for your thoughts, Math Mom. I have a 10 year old and a 14 year old, so I get the study skills point. I believe the study skills issue goes beyond simple adolescence. Most students I’ve encountered professionally rarely study outside of the classroom, or even do homework, whereas most of those I know in my children’s schools complete their homework and study outside the classroom in some fashion. In my classes, those that attempt the homework as well as study score significantly higher than others, however, they are few in number.

      I’ve provided clear notes previously and they did not appear to help. I believe it is related to students not studying from them. As a parent, I know it is difficult to find the time to make sure my kids do their homework, or study for a test, much less help them with it. At the same time, students whose parents minimally make sure students do their homework and study seem to fare better on average. The challenge is ensuring parents are: 1) aware of the need to do so and 2) able to make time to do so, which I can attest, as one of two working parents in our family, is easier said than done.


      • Math Mom says:

        Have you considered providing answer keys? If not for homework then for reviews before the test.

        I go over my son’s math homework. Where there is an error, I circle it and he tries again. He often finds his error or just redoes it correctly, (This is similar to what he does in Kumon but Kumon gives an answer key and for school I have to come up with this myself.)

        This system provides my son with timely feedback. He doesn’t go to sleep thinking his wrong answers are correct. He has also learned to find his errors.

        I would much rather send my son into school knowing he is doing things right than hope for the best – that he is going to be focused enough in a classroom setting to see that he missed a negative sign and fix his incorrect answers. I don’t think he’s ready to do this just yet. Middle school math is critical in being foundations for upper level math. I won’t eave it to chance as to whether he has the right or wrong answers.

        Just an aside – my son has never cheated with the Kumon answer key and can also self-check and make corrections himself. I stay involved with that so I can monitor his

        Many parenta don’t have the time, inclination or skill to do what I do with math homework. That’s why an answer key might be a good help. Sometimes seeing an answer can help you figure out what you’re doing wrong,


  2. Cal says:

    I know your book. I think chapter 2 is a joke. Advanced word problems in chapter 2? The kids just aren’t ready for it. I never taught it for most of my kids, and the advanced ones only got it towards the end of the first semester. So thus far, you’ve spent a long time on concepts, which really isn’t all that connected to the rest of algebra (when will they need to divide a fraction by a fraction?), and extremely difficult word problems. They won’t use any of this again.

    Reteaching is always a bad idea. I would dump inequalities and move onto linear equations.They will need that forever. Most of what you’ve taught them thus far they won’t use again.

    I know I keep saying this, but it bears repeating: consider teaching much, much less than you do.


    • I’m OK with the text, and chapter 2, as a reference, and while the word problems are challenging, they typify the emphasis coming with Common Core, especially related to the standards of mathematical practice, so I plan to continue teaching how to attack these problems to a successful conclusion.

      Beyond number classification, most of the emphasis has been procedural so that students do not get stymied in future work, which requires a basic understanding of operating with integers and rational numbers, as well as combining like terms, evaluating expressions, and solving equations.

      I plan to skip inequalities for now, and cover them later during linear relationships. Subsequent units cover quadratic relationships and exponential relationships, followed by comparing and contrasting those with linear relationships. It is not the typical algebra 1 sequence or scope, and will focus more on problem solving skills than decontextualized exercises.

      Before moving on to linear relationships though, I will reteach the fundamentals covered in test 1. I still believe students must master these essential skills to have any chance of success in algebra 2 or beyond.

      Thanks for your continued support and suggestions!


  3. Dave ( also a career changer Math teacher ),a few years ahead of you. says:

    Having read the previous comments, the test seemed well constructed but perhaps could have had a few less questions ( assuming this was a 45-50 minute class ) as well as maybe less topics. Students seem to confuse different procedures because they have only surface knowledge in many cases. i assume they were using calculators. As an ATR I have seen a myriad of math teaching methodologies. I don’t have the magic answer after 12 years except that if they can’t explain their own ( correct ) work they don’t really know it. I have had the same HW issues; in some cases it is done in such a haphazard, sloppy, and incorrect manner, they might have accomplished more by sleeping.


    • Hi Dave.

      Students had a full 60-minute period for the test. No one asked for more time, which would have been given during lunch or after school. While it was a tad lengthy, I had hoped they would do better as we worked identical problems in class and on quizzes leading up to the test.

      I’m mulling over whether I reteach for the next two to three weeks to see if I can bring the average up to mid sixty-percent range. The alternative is to press on with relationships and functions using multiple representations: verbal, graphical, numerical, and algebraic. My main concern is that skill and understanding deficiencies will get in the way of students attaining proficiency as they will need to handle simplify expressions on their way to solving them algebraically, or write algebraic expressions to reflect graphs, data tables, or word problems.

      I also just realized that hitting the higher average may be more challenging than last year. I advanced students who scored 90% or higher on our department algebra 1 diagnostic test to geometry. This removed the top 10% of students whose scores would have tended towards advanced.


  4. I am curious as to how you reviewed and retaught the material after the first test. Was it in a different manner from the first time through? What were students expected to do during this time?


    • Hi Anna. Given the time crunch of a 3 day reteach, which might be better called a review for a retake, we worked through every problem on the test. During this time, students had their initial tests, and took notes on any problems they did not understand. I also assigned additional homework problems to practice that mirrored the test. We reviewed select problems from the homework, as selected by students.

      Whether I worked a problem, or another student, I always emphasized the key elements of the problem while showing ‘typical’ steps required to address it, to include reading the directions, as many still miss key instructions. I say ‘typical’ as I do not advocate any single method over others procedurally, as most students have already taken algebra at least once. I want students to build on their existing knowledge, or modify it as needed, rather than replace what they learned wholesale.


  5. Pingback: Findings from Frustration | Reflections of a Second-career Math Teacher

  6. eo says:

    It seems like a lot of these questions are the type over which ordinary successful people would stumble. The reason is that these sorts of problems seldom arise in the real world, or when they do they are not in a form designed to confuse. We almost never sort positive and negative numbers together. Few of us evaluate or simplify polynomials, or add fractions — and if we do we have these computer thing … we can type it into google. I think the idea of splitting it into multiple tests, with more questions starting with easy to harder might show some interesting results.

    However, I also wonder if some of the students saw it as a punishment to have to do it over.

    I have a profound dislike for representing the (real) number system with this diagram. I’m not saying the diagram is incorrect, but just not a good way to represent it. This representation seems to put the irrationals outside of the other numbers — when in fact the rationals, integers, etc. are only significant to our human minds, the irrationals are infinitely larger and can do the whole work of the universe without those piddling rationals. Maybe if they were nested boxes touching at the bottom left corner with a dotted line between rational and irrational. In any case you should have a note: boxes not to scale. An other minor point, I think the term natural numbers is pretty old school these days. The wikipedia article, http://en.wikipedia.org/wiki/Natural_number, discuses the fact that there isn’t universal agreement on the meaning of whole and natural numbers. All texts on number theory define their own meaning for counting numbers, rather than assume a universal meaning.


    • Thanks, eo. All students, except one (who scored the highest both times) saw this as an opportunity to improve their scores, which is how it was pitched, and how it was meant.

      While I agree the instances in these problems rarely represent themselves in any day to day situation, they do arise in the need to tease out conceptual understanding or procedural fluency, which is clearly lacking on multiple fronts. Students will need to master these skills to have any chance of attaining high scores in subsequent mathematics courses in high school, or college, for that matter. Additionally, all of the high stakes standardized tests they will see in the next few years (PSAT, SAT, ACT, CAHSEE, CST, ELM) requires they master these skills.

      The point of whether algebra, or above, is ever used in the real world outside of a small subset of mathematicians, statisticians, research scientists, etc. is a valid one. Even as a practicing EE, I rarely worked any mathematics beyond simple addition, subtraction, etc. outside of college or grad school; I did rely heavily on using logarithms to determine link budgets in various communications systems, both satellite and terrestrial. However, the analytical, logical, relational, and topological skills intrinsic in higher order thinking as developed and honed in mathematics coursework is used frequently, albeit in not so obvious ways.

      On the number classification sense, read my post: https://mathequality.wordpress.com/2011/08/02/creating-a-number-set-venn-diagram-poster/

      I really like your tiered test idea. Independently, I have decided to give each student a quick 5-10 minute, 3-5 question quiz on each major section of the test this week as part of our third review leading up to a third chance at the test this Friday. My quizzes should help students self-assess daily, and help me see where they are running into difficulties, which I can address the next day.

      Keep the following in mind, too.

      1) All of the material on this first test covers 4th grade through 7th grade mathematics and was supposed to have been learned, nay mastered, prior to taking algebra 1; no comment on how realistic that might be, however, it is the existing set of expectations embodied in CA standards.

      2) Most of my students took algebra 1 last year, and supposedly used these skills in so doing.

      3) We need to cover significant ground between now and the California State Tests (CSTs) in April, which limits the length of prerequisite review such as this. Also, while I could not “reteach” in any meaningful manner the four years of mathematics spanned in this first unit, I allocated an entire six-week marking period for intensive review as I knew it was necessary before we launched into the Common Core curriculum I plan to trial this year.

      All of these details complicate the pure essence of teaching students mathematics.


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